Five Five Numbers

The number associated with my diurnal age today, \( \textbf{27905} \), is one of those numbers that it is difficult to find anything of interest about. However, as usual, a little investigation turned up something special about it. It is what I call a five five number meaning it meets the following criteria:

• it is a composite and squarefree number
• its digits contain a single 5
• each prime factor contains at least one 5 for a total of three 5's
• its arithmetical digital root is 5
  
  

In the range up to 40000, there are only six numbers that satisfy these criteria and they are 12785, 27635, 27815, 27905, 28265 and 32765. Here are the details (permalink):

 number   factors    root

  12785    5 * 2557   5

  27635    5 * 5527   5

  27815    5 * 5563   5

  27905    5 * 5581   5

  28265    5 * 5653   5

  32765    5 * 6553   5

You could extend this idea to digits other than \( \textbf{5} \). The digit \( \textbf{3} \) produces too many suitable numbers but what about the digit \( \textbf{4} \) where we require this of the number:

• it is a composite and squarefree number
• its digits contain a single 4
• each prime factor contains at least one 4 for a total of two 4's
• its arithmetical digital root is 4
  
  

In the range up to 40000, only one number satisfies and that is 19147 = 41 x 467 with a digital root of 4 (permalink). Nothing for the digit \( \textbf{6} \) up to one million. For the digit \( \textbf{7} \) there is only one number up to one million and that is 544579 = 7 x 77797 with a digital root of 7 - we require that its prime factors contain a total of seven 7's (permalink). For the digits \( \textbf{8} \) and \( \textbf{9} \), no numbers qualify up to one million.

So it turns out that 27905 is not so uninteresting after all. Additionally, it is a \( \textbf{Proth} \) number, since it is equal to \(109 \times 2^8 + 1\) and \(109 < 28 \). I've written about these before in a post titled Proth Numbers.

Triple Seven

Triple seven is often associated with a jackpot when it comes up while playing on poker machines so it's interesting to note when this frequency of sevens occurs in numbers associated with my diurnal age. Today I turned 27377 days old:

How many numbers are there, up to one million let's say, with the property that:

• their digits must contain three 7s
• the number itself is divisible by 7
• the other prime factors have digit sums that are divisible by 7

It turns out that there are only 69 such numbers and they are (permalink):

27377, 28777, 67277, 72737, 77357, 77791, 77917, 79177, 154777, 157787, 172277, 177527, 179767, 197477, 227717, 272797, 280777, 287077, 329777, 347767, 367577, 373877, 447727, 448777, 455777, 477673, 507577, 507787, 644777, 677761, 702737, 702877, 706727, 707791, 717227, 717731, 717857, 720377, 726677, 727517, 727783, 732977, 736757, 737387, 737597, 757379, 760277, 764477, 767473, 770273, 771547, 772037, 774557, 776447, 777091, 777203, 777833, 778337, 779107, 782747, 785771, 791077, 827477, 879277, 896777, 917077, 917707, 977137, 977879

The number associated with my diurnal age, 27377, just happens to be the first of them. I won't list the factorisation of all of the above numbers but I will list those up to 100,000:

• \(27377 = 7 \times 3911\)
• \(28777 = 7 \times 4111\)
• \(67277 = 7^2 \times 1373\)
• \(72737 = 7 \times 10391\)
• \(77357 = 7 \times 43 \times 257\)
• \(77791 = 7 \times 11113\)
• \(77917 = 7 \times 11131\)
• \(79177 = 7\times 11311\)

There are variations possible of course. One could simply require that the number contain three 7s and be divisible by 7. In this case, there are 2070 such numbers in the range up to one million with the smallest of them being 777 and the largest of them being 997787 (permalink). 

Alternatively, we look for numbers containing four 7s instead of three. In this case there are only nine such numbers in the range up to one million and they are 772177, 777217, 777721, 777847, 777973, 778477, 784777, 875777 and 977767 (permalink).

While my focus began with the digit 7 and its threefold repetition within a number, it's easy to modify the earlier algorithm so that it tests for the digit 5. In this case the constraints on the numbers are:

• their digits must contain three 5s
• the number itself is divisible by 5
• the other prime factors have digit sums that are divisible by 5

In the range up to one million there are 426 such numbers, the first being 1555 and the last being 998555 (permalink). As with the 7s, the three 5s do no need to be sequential, although they are sequential in these two examples. Their details are as follows:$$ \begin{align} 1555 &= 5 \times 311 \\ 998555 &= 5 \times 41 \times 4871 \end{align}$$We can only test the digits 5 and 7 using this algorithm. The other prime factors cannot have digit sums that are divisible by 2, 3, 4, 6, 8 or 9 because then they would not be prime and the digits 0 and 1 are obviously excluded.

Digits 3 to 9 in Conway's Game of Life

In my previous post, I looked at the behaviour of the digits 0, 1 and 2 under the rules of Conway's Game of Life. Today I'll look at the digits 3, 4, 5, 6, 7, 8 and 9. Let's start with the digit 3. See Figure 1.

Figure 1: 3 in the shape of an 11-omino

After about 50 steps it ends up in the form shown in Figure 2. There are two ships, two blocks, two beehives and one blinker.

Figure 2: three types of still life and one blinker

Now let's look at the digit 4 shown in Figure 3. It completely disappears after 12 steps or generations, so there's no final state that needs to shown.

Figure 3: the digit 4 in the shape of an octomino
It disappears after 12 generations

The digit 5 is shown in Figure 4 and after three steps or generations it changes into the shapes shown in Figure 5. It's really the same shape as the digit 2 and so the outcomes are basically the same, just differently orientated.

Figure 4: the digit 5 in the shape of an 11-omino

Figure 5: final state of 5 produces two boats

The digit 6 shown in Figure 6 has by far the most complicated behaviour of all the digits. After well over a thousand generations it turns into what is shown in Figure 7.

Figure 6: the digit 6 in the shape of a 12-omino

Figure 7: the complicated final state of the digit 6.
There are additional gliders not shown

Figure 8 shows the digit 7 that, after six generations, turns into a blinker.

Figure 8: the digit 7 as an heptomino
After six generations it becomes a blinker

The digit 8, shown in Figure 9, disappears after 21 generations:

Figure 9: the digit 8 as a 13-omino
It disappears after 21 generations

The digit 9, shown in Figure 10, will behave exactly the same way as for the digit 6, only the orientation will be different.

Figure 10: the digit 9 represented as a 12-omino
It behaves the same as the digit 6